In Neukirch's Algebraic Number Theory, the formulation of Hensel's Lemma (Proposition 4.6 in Chapter II) does not require the valuation to be discrete, only nonarchimedean (unless I somehow missed the requirement).
Following the proof, I used the representation of an element $x \in \mathcal{O}$ as $x = u \pi^n$, which holds when the valuation is discrete.
The Wikipedia article also states the lemma for discrete valuations only.
My question is: It the discreteness of the valuation necessary? If so, is it assumed in the book that every valuation is discrete from this point on?
I have found this question where it is explained that nondiscrete valuations are often omitted in some branches of mathematics, so it would make sense for Neukirch to only consider discrete valuations but again, I have not seen this mentioned.
Best Answer
Was there a step in the proof in Neukirch (which did not assume discreteness) that you did not understand? Note that the $\pi$ in Neukirch's proof is not a choice of prime element in $\mathcal O$. It is a number of largest absolute value among the coefficients in two polynomials whose coefficients are all in the maximal ideal of $\mathcal O$ (so necessarily $|\pi| < 1$). I agree it can look misleading to see powers of a number written as $\pi$, since that suggests $\pi$ is a prime element of $\mathcal O$, but nowhere does he need the maximal ideal of $\mathcal O$ to be generated by $\pi$.
Another book with a formulation of Hensel's lemma in the form you see in Neurkich is Theorem 4.1 in Dwork, Gerotto, and Sullivan's "An Introduction to $G$-Functions". They work in a complete non-archimedean valued field with no assumption of discreteness and their proof is different from the one in Neurkirch, using the contraction mapping theorem on a space of polynomials of bounded degree and no powers of a specially chosen element to create power series in polynomials of bounded degree. Their form of Hensel's lemma is more general than what is in Neukirch: instead of assuming $f \not\equiv 0 \bmod \mathfrak p$ and that there are polynomials $g_0$ and $h_0$ in $\mathcal O[x]$ such that $f \equiv g_0h_0 \bmod \mathfrak p$ where $\gcd(g_0 \bmod \mathfrak p,h_0 \bmod \mathfrak p) = 1$ in $(\mathcal O/\mathfrak p)[x]$, they assume that there are polynomials $g_0$ and $h_0$ in $\mathcal O[x]$ such that$|f - g_0h_0|_{\rm Gauss} < |R(g_0,h_0)|^2$, where $|F|_{\rm Gauss}$ for a polynomial $F$ is the maximal absolute value of the coefficients of $F$. The version of Hensel's lemma in Neukirch is the special case of the version in DGS where $R(g_0,h_0) \not\equiv 0 \bmod \mathfrak p$ (equivalently, $|R(g_0,h_0)|_{\rm Gauss} = 1$). These two versions of Hensel's lemma are analogous to the two standard versions of Hensel's lemma in the formulation about lifting a root: (i) there is $\alpha_0 \in \mathcal O$ such that $f(\alpha_0) \equiv 0 \bmod \mathfrak p$ and $f'(\alpha_0) \not\equiv 0 \bmod \mathfrak p$ compared to (ii) there is $\alpha_0 \in \mathcal O$ such that $|f(\alpha_0)| < |f'(\alpha_0)|^2$, with (i) being the special case where $|f'(\alpha_0)| = 1$. (A root-lifting version of Hensel's lemma is a special case of a factorization-lifting version of Hensel's lemma by taking one polynomial in the factorization to be monic and linear: if $F(x) = (x - \alpha_0)h(x)$ then $R(x-\alpha_0,h(x)) = h(\alpha_0) = F'(\alpha_0)$.)
The proof of Hensel's lemma in Borevich and Shafarevich's "Number Theory" (p. 273) is a factorization-lifting theorem as in Neukirch, but their framework is more restrictive than Neukirch's in one sense (their nonarchimedean absolute value is discrete) and more general than Neukirch's in another sense (their hypothesis involves resultants rather than a relatively prime factorization mod $\mathfrak p$).